Prove matrix power formula

5 Let \(\mathbf { A } = \left( \begin{array} { l l } 1 & a \\ 0 & 1 \end{array} \right)\), where \(a\) is a positive constant.

1. State the type of the geometrical transformation in the \(x - y\) plane represented by \(\mathbf { A }\).
2. Prove by mathematical induction that, for all positive integers \(n\), $$\mathbf { A } ^ { \mathrm { n } } = \left( \begin{array} { c c } 1 & \mathrm { na } \\ 0 & 1 \end{array} \right)$$ Let \(\mathbf { B } = \left( \begin{array} { c c } b & b \\ a ^ { - 1 } & a ^ { - 1 } \end{array} \right)\), where \(b\) is a positive constant.
3. Find the equations of the invariant lines, through the origin, of the transformation in the \(x - y\) plane represented by \(\mathbf { A } ^ { n } \mathbf { B }\).

1 Let \(\mathbf { A } = \left( \begin{array} { l l } 3 & 0 \\ 1 & 1 \end{array} \right)\).

1. Prove by mathematical induction that, for all positive integers \(n\), $$2 \mathbf { A } ^ { n } = \left( \begin{array} { l l }

1. (i) Prove by induction that for \(n \in \mathbb { Z } ^ { + }\)

$$\left( \begin{array} { r r } 5 & - 1 \\ 4 & 1 \end{array} \right) ^ { n } = 3 ^ { n - 1 } \left( \begin{array} { c c } 2 n + 3 & - n \\ 4 n & 3 - 2 n \end{array} \right)$$ (ii) Prove by induction that for \(n \in \mathbb { Z } ^ { + }\) $$f ( n ) = 8 ^ { 2 n + 1 } + 6 ^ { 2 n - 1 }$$ is divisible by 7

1. Prove by induction that, for \(n \in \mathbb { Z } ^ { + }\)

$$\left( \begin{array} { l l } a & 0 \\ 1 & b \end{array} \right) ^ { n } = \left( \begin{array} { c c } a ^ { n } & 0 \\ \frac { a ^ { n } - b ^ { n } } { a - b } & b ^ { n } \end{array} \right)$$ where \(a\) and \(b\) are constants and \(a \neq b\).

1. (a) Prove by induction that for \(n \in \mathbb { Z } ^ { + }\)

$$\left( \begin{array} { l l } 1 & r \\ 0 & 2 \end{array} \right) ^ { n } = \left( \begin{array} { c c } 1 & \left( 2 ^ { n } - 1 \right) r \\ 0 & 2 ^ { n } \end{array} \right)$$ where \(r\) is a constant. $$\mathbf { M } = \left( \begin{array} { l l } 4 & 0 \\ 0 & 5 \end{array} \right) \quad \mathbf { N } = \left( \begin{array} { r r } 1 & - 2 \\ 0 & 2 \end{array} \right) ^ { 4 }$$ The transformation represented by matrix \(\mathbf { M }\) followed by the transformation represented by matrix \(\mathbf { N }\) is represented by the matrix \(\mathbf { B }\) (b) (i) Determine \(\mathbf { N }\) in the form \(\left( \begin{array} { l l } a & b \\ c & d \end{array} \right)\) where \(a , b , c\) and \(d\) are integers.
(ii) Determine B Hexagon \(S\) is transformed onto hexagon \(S ^ { \prime }\) by matrix \(\mathbf { B }\) (c) Given that the area of \(S ^ { \prime }\) is 720 square units, determine the area of \(S\)

9. Prove by induction, that for \(n \in \mathbb { Z } ^ { + }\),

1. \(\left( \begin{array} { l l } 3 & 0 \\ 6 & 1 \end{array} \right) ^ { n } = \left( \begin{array} { c c } 3 ^ { n } & 0 \\ 3 \left( 3 ^ { n } - 1 \right) & 1 \end{array} \right)\),
2. \(\mathrm { f } ( n ) = 7 ^ { 2 n - 1 } + 5\) is divisible by 12 .

1. (i) Prove by induction that, for \(n \in \mathbb { Z } ^ { + }\),

$$\left( \begin{array} { r r } 1 & 0 \\ - 1 & 5 \end{array} \right) ^ { n } = \left( \begin{array} { c c } 1 & 0 \\ - \frac { 1 } { 4 } \left( 5 ^ { n } - 1 \right) & 5 ^ { n } \end{array} \right)$$ (ii) Prove by induction that, for \(n \in \mathbb { Z } ^ { + }\), $$\sum _ { r = 1 } ^ { n } ( 2 r - 1 ) ^ { 2 } = \frac { 1 } { 3 } n \left( 4 n ^ { 2 } - 1 \right)$$

9. Use the method of mathematical induction to prove that, for \(n \in \mathbb { Z } ^ { + }\),

1. \(\left( \begin{array} { c c } 2 & 1 \\ - 1 & 0 \end{array} \right) ^ { n } = \left( \begin{array} { c c } n + 1 & n \\ - n & 1 - n \end{array} \right)\)
2. \(\mathrm { f } ( n ) = 4 ^ { n } + 6 n - 1\) is divisible by 3 .

4 The matrix \(\mathbf { A }\) is given by \(\mathbf { A } = \left( \begin{array} { l l } 3 & 1 \\ 0 & 1 \end{array} \right)\). Prove by induction that, for \(n \geqslant 1\), $$\mathbf { A } ^ { n } = \left( \begin{array} { c c } 3 ^ { n } & \frac { 1 } { 2 } \left( 3 ^ { n } - 1 \right) \\ 0 & 1 \end{array} \right)$$

4 The matrix \(\mathbf { M }\) is given by \(\mathbf { M } = \left( \begin{array} { l l } 2 & 2 \\ 0 & 1 \end{array} \right)\). Prove by induction that, for \(n \geqslant 1\), $$\mathbf { M } ^ { n } = \left( \begin{array} { c c } 2 ^ { n } & 2 ^ { n + 1 } - 2 \\ 0 & 1 \end{array} \right) .$$

2 Let \(\mathbf { A } = \left( \begin{array} { l l } 2 & 3 \\ 0 & 1 \end{array} \right)\). Prove by mathematical induction that, for every positive integer \(n\), $$\mathbf { A } ^ { n } = \left( \begin{array} { c c } 2 ^ { n } & 3 \left( 2 ^ { n } - 1 \right) \\ 0 & 1 \end{array} \right)$$

13 Given that \(\mathbf { M } = \left[ \begin{array} { l l l } 1 & 1 & 1 \\ 1 & 1 & 1 \\ 1 & 1 & 1 \end{array} \right]\), prove that \(\mathbf { M } ^ { n } = \left[ \begin{array} { l l l } 3 ^ { n - 1 } & 3 ^ { n - 1 } & 3 ^ { n - 1 } \\ 3 ^ { n - 1 } & 3 ^ { n - 1 } & 3 ^ { n - 1 } \\ 3 ^ { n - 1 } & 3 ^ { n - 1 } & 3 ^ { n - 1 } \end{array} \right]\) for all \(n \in \mathbb { N }\) [0pt] [5 marks] LL LL L
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6 You are given that \(\mathbf { A } = \left( \begin{array} { l l } 1 & a \\ 0 & 1 \end{array} \right)\) where \(a\) is a constant.
Prove by induction that \(\mathbf { A } ^ { \mathrm { n } } = \left( \begin{array} { c c } 1 & \text { an } \\ 0 & 1 \end{array} \right)\) for all integers \(n \geqslant 1\).

8 Prove by induction that \(\left( \begin{array} { l l } 1 & 1 \\ 0 & 2 \end{array} \right) ^ { n } = \left( \begin{array} { c c } 1 & 2 ^ { n } - 1 \\ 0 & 2 ^ { n } \end{array} \right)\) for all positive integers \(n\).

7. Using mathematical induction, prove that $$\left[ \begin{array} { l l } 2 & 5 \\ 0 & 2 \end{array} \right] ^ { n } = \left[ \begin{array} { c c } 2 ^ { n } & 2 ^ { n - 1 } \times 5 n \\ 0 & 2 ^ { n } \end{array} \right]$$ for all positive integers \(n\).

1. (i) Prove by induction that for \(n \in \mathbb { Z } ^ { + }\)

$$\left( \begin{array} { l l } 5 & - 8 \\ 2 & - 3 \end{array} \right) ^ { n } = \left( \begin{array} { c c } 4 n + 1 & - 8 n \\ 2 n & 1 - 4 n \end{array} \right)$$ (ii) Prove by induction that for \(n \in \mathbb { Z } ^ { + }\) $$f ( n ) = 4 ^ { n + 1 } + 5 ^ { 2 n - 1 }$$ is divisible by 21

1. Prove by mathematical induction that, for \(n \in \mathbb { N }\)

$$\left( \begin{array} { l l } - 5 & 9 \\ - 4 & 7 \end{array} \right) ^ { n } = \left( \begin{array} { c c } 1 - 6 n & 9 n \\ - 4 n & 1 + 6 n \end{array} \right)$$

1. Prove by induction that for \(n \in \mathbb { N }\)

$$\left( \begin{array} { c c } 1 & - 2 \\ 0 & 1 \end{array} \right) ^ { n } = \left( \begin{array} { c c } 1 & - 2 n \\ 0 & 1 \end{array} \right)$$

5 The matrix \(\mathbf { M }\) is defined by \(\mathbf { M } = \left[ \begin{array} { c c c } 3 & 2 & - 2 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array} \right]\) Prove by induction that \(\mathbf { M } ^ { n } = \left[ \begin{array} { c c c } 3 ^ { n } & 3 ^ { n } - 1 & - 3 ^ { n } + 1 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array} \right]\) for all integers \(n \geq 1\) [5 marks]